When you say 1080i HDTV, do you mean live digital TV? Because I would think that your tuner card, drivers and codec choice would be important considerations as much as anything?
For media playback I found that each release of the Intel drivers was improving the situation, but the picture quality was still crap on all high-def material. Kind of like it was struggling with contrast or colour depth, + some blockiness. The posts by drealit on page 1 give you the idea of symptoms and cause. Even as of December the situation had improved (see post by pgroe on page 2), but still lagged behind ATI and nVidia.
I gave up, purchased an ATI HD5450 and never looked back. The Core i3/5 series have plenty of power for a HTPC with good thermal properties and are pretty cost-effective. But my advice is that if you value your time and care about picture quality, spend $40 on a discrete video card like an ATI 5450 or G210...
Yes, I mean live TV, from satellite. I guess I don't have anything to compare it to, but I'd never describe the PQ as "crap" - it's certainly (subjectively) better than my S*y HD box, and that's good for me. And yes the tuner card drivers and codec are indeed all important, but the point for me was that the previous Intel on-board GPU I was using wasn't quite grunty enough to play HDTV properly in MP - the one I have now (Core i3) is.
(NB I had issues the one time I tried an nvidia card, in that it sent my TV's power-saving feature crazy so the picture was dimming and un-dimming every few seconds - so it was no good for me. Never tried ATI, admittedly.)
I've been running an i3 setup for nearly a near now and I have ZERO complaints with it's performance. I have no issue with codecs and to me the PQ is excellent. I'm using the Microsoft DTV codecs for all the video decoders and ffdshow for all the audio. Works perfectly fine for me regardless of what I throw at it. 1080i ATSC Live TV, Blu-ray, h264, VC-1, you name it. It just works and it was far easier to set up and get working than anything from ATI or nVidia that I've ever used.